Proof. As M is an n-dimensional complex manifold satisfying condition $(a)$ and $c(t)\in \mathcal {P}_T$ , there exist a closed subset $X\subset M$ and a closed subset $E\subset X\cap \{\psi =-\infty \}$ satisfying that X is locally negligible with respect to $L^2$ holomorphic functions, $M\backslash X$ is a Stein manifold, $e^{-\varphi }c(-\psi )$ has locally a positive lower bound on $M\backslash E$ and there exists an analytic subset Z of M such that $E\subset Z$ .

Combining inequality (10) and $e^{-\varphi }c(-\psi )$ has locally a positive lower bound on $M\backslash E$ , we obtain that

$$ \begin{align*}\int_{K\cap\{\psi<-t_{0}\}}|F|^{2}<+\infty\end{align*} $$

holds for any compact subset K of $M\backslash X$ . Then Lemma 2.1 shows that there exists a holomorphic $(n,0)$ form $\tilde {F}_X$ on $\{\psi <-t_1\}\backslash X$ , such that

(12) $$ \begin{align} \int_{\{\psi<-t_1\}\backslash X}|\tilde{F}_X-(1-b_{t_0,B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\leq C\int_{t_1}^{t_{0}+B}c(t)e^{-t}dt. \end{align} $$

For any $z\in \{\psi <-t_1\}\cap (X\backslash E)$ , there exists an open neighborhood $V_z$ of z, such that $V_z\subset \subset \{\psi <-t_1\}\backslash E$ . Note that $c(t)e^{-t}$ is decreasing on $(T,+\infty )$ and $v_{t_0,B}(\psi )\geq \psi $ , then we have

(13) $$ \begin{align}\begin{aligned} &\int_{V_z\backslash X}|\tilde{F}_X-(1-b_{t_0,B}(\psi))F|^2e^{-\varphi}c(-\psi)\\ \leq&\int_{V_z\backslash X}|\tilde{F}_X-(1-b_{t_0,B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\\ <&+\infty. \end{aligned} \end{align} $$

Note that there exists a positive number $C_1>0$ such that $e^{-\varphi }c(-\psi )>C_1$ on $V_{z}$ and $\int _{V_z\backslash X}|(1-b_{t_0,B}(\psi ))F|^2e^{-\varphi }c(-\psi )\leq \int _{\{\psi <-t_{0}\}}|F|^{2}e^{-\varphi }c(-\psi )<+\infty $ , then we have

(14) $$ \begin{align} \begin{aligned} &\int_{V_z\backslash X}|\tilde{F}_X|^2\\ \leq &C_1\int_{V_z\backslash X}|\tilde{F}_X|^2e^{-\varphi}c(-\psi) \\ \leq &2C_1\left(\int_{V_z\backslash X}|(1-b_{t_0,B}(\psi))F|^2e^{-\varphi}c(-\psi)+\int_{V_z\backslash X}|\tilde{F}_X-(1-b_{t_0,B}(\psi))F|^2e^{-\varphi}c(-\psi)\right)\\ <&+\infty.\end{aligned} \end{align} $$

As X is locally negligible with respect to $L^2$ holomorphic functions, we can find a holomorphic extension $\tilde {F}_E$ of $\tilde {F}_X$ from $\{\psi <-t_1\}\backslash X$ to $\{\psi <-t_1\}\backslash E$ such that

(15) $$ \begin{align} \int_{\{\psi<-t_1\}\backslash E}|\tilde{F}_E-(1-b_{t_0,B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\leq C\int_{t_1}^{t_{0}+B}c(t)e^{-t}dt. \end{align} $$

Note that $E\subset \{\psi <-t_0\}\subset \{\psi <-t_1\}$ , for any $z\in E$ , there exists an open neighborhood $U_z$ of z, such that $U_z\subset \subset \{\psi <-t_0\}$ . As $\varphi +\psi $ is plurisubharmonic on M and $e^{v_{t_0,B}(\psi )}c(-v_{t_0,B}(\psi ))$ has a positive lower bound on $\{\psi <-t_1\}$ , then we have

(16) $$ \begin{align} \begin{aligned} &\int_{U_z\backslash E}|\tilde{F}_E-(1-b_{t_0,B}(\psi))F|^{2}\\ \leq&C_2\int_{\{\psi<-t_1\}\backslash E}|\tilde{F}_E-(1-b_{t_0,B}(\psi)) F|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\\ <&+\infty, \end{aligned} \end{align} $$

where $C_2$ is some positive number. As $U_z\subset \subset \{\psi <-t_0\}$ , we have

(17) $$ \begin{align} \int_{U_z}|(1-b_{t_0,B}(\psi))F|^2\le\int_{U_z}|F|^2<+\infty. \end{align} $$

Combining inequality (16) and (17), we obtain that $\int _{U_z\backslash E}|\tilde {F}_E|^2<+\infty $ .

As E is contained in some analytic subset of M, we can find a holomorphic extension $\tilde {F}$ of $\tilde {F}_E$ from $\{\psi <-t_1\}\backslash E$ to $\{\psi <-t_1\}$ such that

(18) $$ \begin{align} \int_{\{\psi<-t_1\}}|\tilde{F}-(1-b_{t_0,B}(\psi))F|^{2}e^{-\varphi-\psi+v_{t_0,B}(\psi)}c(-v_{t_0,B}(\psi))\leq C\int_{t_1}^{t_{0}+B}c(t)e^{-t}dt.\end{align} $$

This proves Lemma 2.2.

Proof. As S is a analytic subset of M, by Local Parameterization Theorem (see [Reference Demailly5]) and Maximum Principle, for any compact set $K\subset \subset M$ , there exists $ K_1\subset \subset M\backslash S$ satisfying

(19) $$ \begin{align} \sup_{z\in K}\frac{|F_j(z)|^2}{dV_M}\leq C_1\sup_{z\in K_1}\frac{|F_j(z)|^2}{dV_M} \end{align} $$

for any j, where $C_1$ is a constant depending on K but independent of j. Then there exists a compact set $K_2\subset \subset M\backslash S$ satisfying $K_1\subset K_2$ and

(20) $$ \begin{align} \begin{aligned} \left(\frac{|F_j(z)|^{2}}{dV_M}\right)^r&\leq C_2\int_{K_2}\left(\frac{|F_j(z)|^{2}}{dV_M}\right)^r\\ &\leq C_2 \left(\int_{K_2}|F_j|^{2}g_j\right)^{r}\left(\int_{K_2}g_j^{-\frac{r}{1-r}}\right)^{1-r} \end{aligned}\end{align} $$

for any j and any $z\in K_1$ , where $r\in (0,1)$ and $C_2$ is a constant. Let $\frac {r}{1-r}=s_{K_2}$ , inequality (20) implies

(21) $$ \begin{align} \sup_{z\in K_1}\frac{|F_j(z)|^2}{dV_M}\leq C_3\int_{K_2}|F_j|^{2}g_j, \end{align} $$

where $C_3$ is a constant. As $\liminf _{j\rightarrow +\infty }\int _{M}|F_j|^2g_j<C$ , combining inequality (19), (21), and the diagonal method, we obtain a subsequence of $\{F_j\}$ , denoted still by $\{F_j\}$ , which is uniformly convergent to a holomorphic $(n,0)$ form F on M on any compact subset of M.

It follows from the Fatou’s Lemma and $\lim _{j\rightarrow +\infty }\int _{M}|F_j|^2g_j\leq C$ that

$$ \begin{align*} \int_{M}|F|^2g = & \int_{M}\lim_{j\rightarrow+\infty}|F_j|^2g_j\\ \leq & \liminf_{j\rightarrow+\infty}\int_{M}|F_j|^2g_j\\ \leq & C. \end{align*} $$

Thus Lemma 2.4 holds.

Proof. It is clear that $f\in H^0(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})\Rightarrow G(t)=0$ .

In the following part, we prove that $G(t)=0\Rightarrow f\in H^0(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ . As $G(t)=0$ , then there exists holomorphic $(n,0)$ forms $\{\tilde {f}_{j}\}_{j\in \mathbb {N}^{+}}$ on $\{\psi <-t\}$ such that $\lim _{j\to +\infty }\int _{\{\psi <-t\}}|\tilde {f}_{j}|^{2}e^{-\varphi }c(-\psi )=0$ and $(f_{j}-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ for any j. As $e^{-\varphi }c(-\psi )$ has a positive lower bound on any compact subset of $M\backslash Z$ , where Z is some analytic subset of M, it follows from Lemma 2.4 that there exists a subsequence of $\{\tilde {f}_{j}\}_{j\in \mathbb {N}^{+}}$ denoted by $\{\tilde {f}_{j_{k}}\}_{k\in \mathbb {N}^{+}}$ that compactly convergent to $0$ . It is clear that $\tilde {f}_{j_{k}}-f$ is compactly convergent to $0-f=-f$ on $U\cap \{\psi <-t\}$ . It follows from Lemma 2.3 that $f\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ . This proves Lemma 2.5.

Proof. Firstly, we prove the existence of $F_{t}$ . As $G(t)<+\infty $ then there exists holomorphic $(n,0)$ forms $\{f_{j}\}_{j\in \mathbb {N}^{+}}$ on $\{\psi <-t\}$ such that $\lim _{j\rightarrow +\infty }\int _{\{\psi <-t\}}|f_{j}|^{2}e^{-\varphi }c(-\psi )=G(t)$ , and $(f_{j}-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ . As $e^{-\varphi }c(-\psi )$ has a positive lower bound on any compact subset of $M\backslash Z$ , where Z is some analytic subset of M, it follows from Lemma 2.4 that there exists a subsequence of $\{f_{j}\}$ compact convergence to a holomorphic $(n,0)$ form F on $\{\psi <-t\}$ satisfying $ \int _{\{\psi <-t\}}|F|^{2}e^{-\varphi }c(-\psi )\leq G(t)$ . It follows from Lemma 2.3 that $(F-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ . Then we obtain the existence of $F_{t}(=F)$ .

Secondly, we prove the uniqueness of $F_{t}$ by contradiction: if not, there exist two different holomorphic $(n,0)$ forms $f_{1}$ and $f_{2}$ on on $\{\psi <-t\}$ satisfying $\int _{\{\psi <-t\}}|f_{1}|^{2}e^{-\varphi }c(-\psi )=\int _{\{\psi <-t\}}|f_{2}|^{2}=G(t)$ , $(f_{1}-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and $(f_{2}-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ . Note that

(23) $$ \begin{align} \begin{aligned} &\int_{\{\psi<-t\}}\left|\frac{f_{1}+f_{2}}{2}\right|{}^{2}e^{-\varphi}c(-\psi)+\int_{\{\psi<-t\}}\left|\frac{f_{1}-f_{2}}{2}\right|{}^{2}e^{-\varphi}c(-\psi) \\=& \frac{\int_{\{\psi<-t\}}|f_{1}|^{2}e^{-\varphi}c(-\psi)+\int_{\{\psi<-t\}}|f_{2}|^{2}e^{-\varphi}c(-\psi)}{2}=G(t), \end{aligned} \end{align} $$

then we obtain that

$$ \begin{align*}\int_{\{\psi<-t\}}\left|\frac{f_{1}+f_{2}}{2}\right|{}^{2}e^{-\varphi}c(-\psi)<G(t),\end{align*} $$

and $(\frac {f_{1}+f_{2}}{2}-f)\in H^{0}(Z_0, (\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ , which contradicts the definition of $G(t)$ .

Finally, we prove equality (22). For any holomorphic h on $\{\psi <-t\}$ satisfying $\int _{\{\psi <-t\}}|h|^{2}e^{-\varphi }c(-\psi )<+\infty $ and $h\in H^{0}(Z_0, (\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ , it is clear that for any complex number $\alpha $ , $F_{t}+\alpha h$ satisfying $((F_{t}+\alpha h)-f)\in H^{0}(Z_0, (\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ , and $\int _{\{\psi <-t\}}|F_{t}|^{2}e^{-\varphi }c(-\psi )\leq \int _{\{\psi <-t\}}|F_{t}+\alpha h|^{2}e^{-\varphi }c(-\psi )<+\infty $ . Note that

$$ \begin{align*}\int_{\{\psi<-t\}}|F_{t}+\alpha h|^{2}e^{-\varphi}c(-\psi)-\int_{\{\psi<-t\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)\geq 0\end{align*} $$

implies

$$ \begin{align*}\Re\int_{\{\psi<-t\}}F_{t}\bar{h}e^{-\varphi}c(-\psi)=0,\end{align*} $$

then

$$ \begin{align*}\int_{\{\psi<-t\}}|F_{t}+h|^{2}e^{-\varphi}c(-\psi)=\int_{\{\psi<-t\}}(|F_{t}|^{2}+|h|^{2})e^{-\varphi}c(-\psi).\end{align*} $$

Choosing $h=\hat {F}-F_{t}$ , we obtain equality (22).

Proof. By the definition of $G(t)$ , it is clear that $G(t)$ is decreasing on $[T,+\infty )$ . And it follows from the dominated convergence theorem that if $G(t)<+\infty $ for some $t\geq T$ , then $\lim _{t\to +\infty }G(t)=0$ . Then it suffices to prove $\lim _{t\to t_{0}+0}G(t)=G(t_{0}).$ We prove it by contradiction: if not, then $\lim _{t\to t_{0}+0}G(t)<G(t_{0})$ .

By Lemma 2.6, there exists a unique holomorphic $(n,0)$ form $F_{t}$ on $\{\psi <-t\}$ satisfying $(F_{t}-f)\in H^{0}(Z_0, (\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and $\int _{\{\psi <-t\}}|F_{t}|^{2}e^{-\varphi }c(-\psi )=G(t)$ . Note that $G(t)$ is decreasing implies that $\int _{\{\psi <-t\}}|F_{t}|^{2}e^{-\varphi }c(-\psi )\leq \lim _{t\to t_{0}+0}G(t)$ for any $t>t_{0}$ . If $\lim _{t\to t_{0}+0}G(t)=+\infty $ , the equality $\lim _{t\to t_{0}+0}G(t)=G(t_{0})$ is clear, thus it suffices to prove the case $\lim _{t\to t_{0}+0}G(t)<+\infty $ . As $e^{-\varphi }c(-\psi )$ has a positive lower bound on any compact subset of $M\backslash Z$ , where Z is some analytic subset of M, and $\int _{\{\psi <-t_1\}}|F_t|^2e^{-\varphi }c(-\psi )\leq \lim _{t\to t_{0}+0}G(t)<+\infty $ holds for any $t\in (t_0,t_1]$ , where $t_1>t_0$ is a fixed number, it follows from Lemma 2.4 that there exists $\{F_{t_{j}}\} \ (t_{j}\to t_{0}+0,$ as $j\to +\infty )$ uniformly convergent on any compact subset of $\{\psi <-t_1\}$ . Using the diagonal method, we obtain a subsequence of $\{F_{t}\}$ (also denoted by $\{F_{t_j}\}$ ), which is convergent on any compact subset of $\{\psi <-t_{0}\}$ .

Let $\hat {F}_{t_{0}}=\lim _{j\to +\infty }F_{t_{j}}$ , which is a holomorphic $(n,0)$ form on $\{\psi <-t_{0}\}$ . Then it follows from the decreasing property of $G(t)$ that

$$ \begin{align*}\int_{K}|\hat{F}_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\leq \lim_{j\to+\infty}\int_{K}|F_{t_{j}}|^{2}e^{-\varphi}c(-\psi)\leq \lim_{j\to+\infty}G(t_{j})\leq\lim_{t\to t_{0}+0}G(t)\end{align*} $$

for any compact set $K\subset \{\psi <-t_{0}\}$ . It follows from Levi’s Theorem that

$$ \begin{align*}\int_{\{\psi<-t_0\}}|\hat{F}_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\leq \lim_{t\to t_{0}+0}G(t).\end{align*} $$

It follows from Lemma 2.3 that $\hat {F}_{t_0}\in H^{0}(Z_0, (\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ . Then we obtain that $G(t_0)\leq \int _{\{\psi <-t_0\}}|\hat {F}_{t_{0}}|^{2}e^{-\varphi }c(-\psi )\leq \lim _{t\to t_{0}+0}G(t)$ , which contradicts $\lim _{t\to t_{0}+0}G(t)<G(t_{0})$ .

Proof. It follows from Lemma 2.7 that $G(t)<+\infty $ for any $t\geq t_1$ . By Lemma 2.6, there exists a holomorphic $(n,0)$ form $F_{t_0}$ on $\{\psi <-t_0\}$ , such that $(F_{t_0}-f)\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and $\int _{\{\psi <-t_0\}}|F_{t_0}|^{2}e^{-\varphi }c(-\psi )=G(t_0)$ .

It suffices to consider that $\liminf _{B\to 0+0}\frac {G(t_0)-G(t_0+B)}{\int _{t_0}^{t_0+B}c(t)e^{-t}dt}\in [0.+\infty )$ because of the decreasing property of $G(t)$ . Then there exists $B_{j}\to 0+0 \ (j\to +\infty )$ such that

(24) $$ \begin{align} \lim_{j\to+\infty}\frac{G(t_0)-G(t_0+B_j)}{\int_{t_0}^{t_0+B_j}c(t)e^{-t}dt}=\liminf_{B\to0+0}\frac{G(t_0)-G(t_0+B)}{\int_{t_0}^{t_0+B}c(t)e^{-t}dt} \end{align} $$

and $\{\frac {G(t_0)-G(t_0+B_j)}{\int _{t_0}^{t_0+B_j}c(t)e^{-t}dt}\}_{j\in \mathbb {N}^{+}}$ is bounded. As $c(t)e^{-t}$ is decreasing and positive on $(T,+\infty )$ , then

(25) $$ \begin{align} \begin{aligned} \lim_{j\to+\infty}\frac{G(t_0)-G(t_0+B_j)}{\int_{t_0}^{t_0+B_j}c(t)e^{-t}dt}=&\left(\lim_{j\to+\infty}\frac{G(t_0)-G(t_0+B_j)}{B_j}\right)\left(\frac{1}{\lim_{t\rightarrow t_0+0}c(t)e^{-t}}\right)\\ =&\left(\lim_{j\to+\infty}\frac{G(t_{0})-G(t_{0}+B_j)}{B_j}\right)\left(\frac{e^{t_0}}{\lim_{t\rightarrow t_0+0}c(t)}\right). \end{aligned} \end{align} $$

Hence, $\left \{\frac {G(t_0)-G(t_0+B_j)}{B_j}\right \}_{j\in \mathbb {N}^{+}}$ is bounded with respect to j.

As $t\leq v_{t_0,B_j}(t)$ , the decreasing property of $c(t)e^{-t}$ shows that

$$ \begin{align*}e^{-\psi+v_{t_0,B_j}(\psi)}c(-v_{t_0,B_j}(\psi))\geq c(-\psi).\end{align*} $$

Lemma 2.2 shows that for any $B_{j}$ , there exists holomorphic $(n,0)$ form $\tilde {F}_{j}$ on $\{\psi <-t_1\}$ , such that $(\tilde {F}_{j}-F_{t_{0}})\in H^{0}(Z_0,(\mathcal {O}(K_M)\otimes \mathcal {I}(\varphi +\psi ))|_{Z_0}) \subseteq H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ and

(26) $$ \begin{align} \begin{aligned} &\int_{\{\psi<-t_1\}}|\tilde{F}_{j}-(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi) \\\leq&\int_{\{\psi<-t_1\}}|\tilde{F}_{j}-(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}e^{-\psi+v_{t_0,B_j}(\psi)}c(-v_{t_0,B_j}(\psi)) \\\leq& \int_{t_1}^{t_{0}+B_{j}}c(t)e^{-t}dt \int_{\{\psi<-t_1\}}\frac{1}{B_{j}}(\mathbb{I}_{\{-t_{0}-B_{j}<\psi<-t_{0}\}})|F_{t_{0}}|^{2}e^{-\varphi-\psi} \\\leq& \frac{e^{t_{0}+B_{j}}\int_{t_1}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)} \int_{\{\psi<-t_1\}}\frac{1}{B_{j}}(\mathbb{I}_{\{-t_{0}-B_{j}<\psi<-t_{0}\}})|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi) \\\leq& \frac{e^{t_{0}+B_{j}}\int_{t_1}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)} \times\Bigg(\int_{\{\psi<-t_1\}}\frac{1}{B_{j}}\mathbb{I}_{\{\psi<-t_{0}\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi) \\&-\int_{\{\psi<-t_1\}}\frac{1}{B_{j}}\mathbb{I}_{\{\psi<-t_{0}-B_{j}\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\Bigg) \\\leq& \frac{e^{t_{0}+B_{j}}\int_{t_1}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)} \times\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}. \end{aligned} \end{align} $$

Firstly, we will prove that $\int _{\{\psi <-t_1\}}|\tilde {F}_{j}|^{2}e^{-\varphi }c(-\psi )$ is bounded with respect to j.

Note that

(27) $$ \begin{align} \begin{aligned} &\left(\int_{\{\psi<-t_1\}}|\tilde{F}_{j}-(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}} \\ \geq&\left(\int_{\{\psi<-t_1\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}-\left(\int_{\{\psi<-t_1\}}|(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}, \end{aligned} \end{align} $$

then it follows from inequality (26) that

(28) $$ \begin{align} \begin{aligned} &\left(\int_{\{\psi<-t_1\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}} \\\leq&\left(\frac{e^{t_{0}+B_{j}}\int_{t_1}^{t_{0}+B_{j}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j})}c(t)}\right)^{\frac{1}{2}} \left(\frac{G(t_{0})-G(t_{0}+B_{j})}{B_{j}}\right)^{\frac{1}{2}} \\&+\left(\int_{\{\psi<-t_1\}}|(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\right)^{\frac{1}{2}}. \end{aligned} \end{align} $$

Since $\left \{\frac {G(t_{0}+B_{j})-G(t_{0})}{B_{j}}\right \}_{j\in \mathbb {N}^{+}}$ is bounded, $\lim _{j\rightarrow +\infty }\inf _{t\in (t_{0},t_{0}+B_{j})}c(t)\in (0,+\infty )$ and

$$ \begin{align*}\int_{\{\psi<-t_1\}}|(1-b_{t_{0},B_{j}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)\leq\int_{\{\psi<-t_0\}}|F_{t_{0}}|^{2}e^{-\varphi}c(-\psi)<+\infty,\end{align*} $$

then $\int _{\{\psi <-t_1\}}|\tilde {F}_{j}|^{2}e^{-\varphi }c(-\psi )$ is bounded with respect to j.

Secondly, we will prove the main result.

It follows from $\int _{\{\psi <-t_1\}}|\tilde {F}_{j}|^{2}e^{-\varphi }c(-\psi )$ is bounded with respect to j and Lemma 2.4 that there exists a subsequence of $\{\tilde {F}_j\}$ , denoted by $\{\tilde {F}_{j_k}\}_{k\in \mathbb {N}^+}$ , which is uniformly convergent to a holomorphic $(n,0)$ form $F_1$ on $\{\psi <-t_1\}$ on any compact subset of $\{\psi <-t_1\}$ when $k\rightarrow +\infty $ , such that

$$ \begin{align*}\int_{\{\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)\le\liminf_{j\rightarrow+\infty}\int_{\{\psi<-t_1\}}|\tilde{F}_{j}|^{2}e^{-\varphi}c(-\psi)<+\infty.\end{align*} $$

As $(\tilde {F}_j-F_{t_{0}})\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ for any j, we have $(F_1-F_{t_{0}})\in H^{0}(Z_0,(\mathcal {O}(K_{M})\otimes \mathcal {F})|_{Z_0})$ . Note that

$$ \begin{align*} \lim_{j\rightarrow+\infty}b_{t_0,B_j}(t)=\lim_{j\rightarrow+\infty}\int_{-\infty}^{t}\frac{1}{B_j}\mathbb{I}_{\{-t_0-B_j<s<-t_0\}}ds=\left\{ \begin{array}{lcl} 0, & \mbox{if}& x\in(-\infty,-t_0),\\ 1, & \mbox{if}& x\in[-t_0,+\infty), \end{array} \right. \end{align*} $$

and

$$ \begin{align*} \lim_{j\rightarrow+\infty}v_{t_0,B_j}(t)=\lim_{j\rightarrow+\infty}\int_{-t_0}^{t}b_{t_0,B_j}ds-t_0=\left\{ \begin{array}{lcl} -t_0, & \mbox{if}& x\in(-\infty,-t_0),\\ t, & \mbox{if}& x\in[-t_0,+\infty). \end{array} \right. \end{align*} $$

Following from equality (25), inequality (26), and the Fatou’s Lemma, we have

(29) $$ \begin{align} \begin{aligned} &\int_{\{\psi<-t_0\}}|F_1-F_{t_0}|^2e^{-\varphi-\psi-t_0}c(t_0)+\int_{\{-t_0\le\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)\\ =&\int_{\{\psi<-t_1\}}\lim_{k\rightarrow+\infty}|\tilde{F}_{j_k}-(1-b_{t_{0},B_{j_k}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}e^{-\psi+v_{t_0,B_{j_k}}(\psi)}c(-v_{t_0,B_{j_k}}(\psi)) \\ \le&\liminf_{k\rightarrow+\infty}\int_{\{\psi<-t_1\}}|\tilde{F}_{j_k}-(1-b_{t_{0},B_{j_k}}(\psi))F_{t_{0}}|^{2}e^{-\varphi}e^{-\psi+v_{t_0,B_{j_k}}(\psi)}c(-v_{t_0,B_{j_k}}(\psi)) \\ \leq& \liminf_{k\rightarrow+\infty}\left(\frac{e^{t_{0}+B_{j_k}}\int_{t_1}^{t_{0}+B_{j_k}}c(t)e^{-t}dt}{\inf_{t\in(t_{0},t_{0}+B_{j_k})}c(t)} \times\frac{G(t_{0})-G(t_{0}+B_{j_k})}{B_{j_k}}\right)\\ =&\frac{e^{t_0}\int_{t_1}^{t_0}c(t)e^{-t}dt}{\lim_{t\rightarrow t_0+0}c(t)}\lim_{j\rightarrow +\infty}\frac{G(t_0)-G(t_0+B_j)}{B_j}\\ =&\int_{t_1}^{t_0}c(t)e^{-t}dt\lim_{j\rightarrow +\infty}\frac{G(t_0)-G(t_0+B_j)}{\int_{t_0}^{t_0+B_j}c(t)e^{-t}dt}. \end{aligned} \end{align} $$

As $e^{\psi }c(-\psi )\le e^{-t_0}c(t_0)$ on $\{\psi <-t_0\}$ , it follows Lemma 2.6, equality (24) and inequality (29) that

(30) $$ \begin{align} \begin{aligned} &\int_{t_1}^{t_0}c(t)e^{-t}dt\liminf_{B\rightarrow 0+0}\frac{G(t_0)-G(t_0+B)}{\int_{t_0}^{t_0+B}c(t)e^{-t}dt}\\ =&\int_{t_1}^{t_0}c(t)e^{-t}dt\lim_{j\rightarrow +\infty} \frac{G(t_0)-G(t_0+B_j)}{\int_{t_0}^{t_0+B_j}c(t)e^{-t}dt}\\ \ge& \int_{\{\psi<-t_0\}}|F_1-F_{t_0}|e^{-\varphi-\psi-t_0}c(t_0)+\int_{\{-t_0\le\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)\\ \ge&\int_{\{\psi<-t_0\}}|F_1-F_{t_0}|e^{-\varphi}c(-\psi)+\int_{\{-t_0\le\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)\\ =&\int_{\{\psi<-t_1\}}|F_1|^2e^{-\varphi}c(-\psi)-\int_{\{\psi<-t_0\}}|F_{t_0}|^2e^{-\varphi}c(-\psi)\\ \ge& G(t_1)-G(t_0). \end{aligned} \end{align} $$

This proves Lemma 2.8.

Proof. For the convenience of the reader, we recall the proof.

It suffices to prove that inequality (31) implies the concavity of $a(r)$ . We prove by contradiction: if not, there exists $A< r_3<r_4<r_5<B$ such that

(32) $$ \begin{align} \frac{a(r_4)-a(r_3)}{r_4-r_3}<\frac{a(r_5)-a(r_3)}{r_5-r_3}<\frac{a(r_5)-a(r_4)}{r_5-r_4}. \end{align} $$

Consider $\tilde a(r)=a(r)-a(r_5)-\frac {a(r_5)-a(r_3)}{r_5-r_3}(r-r_5)$ on $(A,B)$ . As $a(r)$ is lower semicontinuous on $(A,B)$ , then $\tilde a(r)$ is lower semicontinuous on $(A,B)$ . Note that $\tilde {a}(r_3)=\tilde {a}(r_5)=0$ and $\tilde {a}(r_4)<0$ , then it follows from the lower semicontinuity of $\tilde {a}(r)$ that there exists $r_6\in (r_3,r_5)$ such that $\tilde {a}(r_6)=\inf _{r\in [r_3,r_5]}\tilde {a}(r)<0$ . It clear that $\frac {\tilde {a}(r_6)-\tilde {a}(r_3)}{r_6-r_3}<0$ and $\limsup _{r\rightarrow r_6+0}\frac {\tilde {a}(r)-\tilde {a}(r_6)}{r-r_6}\geq 0$ . Then we obtain that

$$ \begin{align*}\frac{{a}(r_6)-{a}(r_3)}{r_6-r_3}<\frac{a(r_5)-a(r_3)}{r_5-r_3}\leq\limsup_{r\rightarrow r_6+0}\frac{{a}(r)-{a}(r_6)}{r-r_6},\end{align*} $$

which contradict inequality (31).

Proof. Without loss of generality, we can assume that $T<0$ . Let $a_n=c(n)e^{-n}$ , where $n\in \mathbb {N}^+$ . Take $b_1=a_1$ , and we can define $b_n=\max {\{\frac {b_{n-1}}{e},a_n\}}$ for $n>1$ , inductively. Since $a_n$ is decreasing with respect to n, we have $b_n\geq b_{n+1}\geq \frac {b_{n}}{e}$ and $b_n\geq a_n$ for any $n\in \mathbb {N}^{+}$ .

Let

$$ \begin{align*} \tilde a(t)=\left\{\begin{array}{lcl} eb_n(\frac{b_{n+1}}{b_n})^{t-n}, & \mbox{if} & t\in[n,n+1),\\ c(t)e^{-t+1}, &\mbox{if} & t\in(T,1]. \end{array} \right. \end{align*} $$

It is clear that $\tilde a(t)\geq c(t)e^{-t}$ , $\tilde a(t)$ is decreasing on $(T,+\infty )$ and continuous on $[1,+\infty )$ . Let $\tilde c(t)=\tilde a(t)e^{t}$ . When $t\in [n,n+1)$ , as $eb_{n+1}\geq b_n$ , we have $\tilde c(t)$ is increasing on $[n,n+1)$ , which implies that $\tilde c(t)$ is increasing on $(1,+\infty )$ .

As $\int _{0}^{+\infty }c(t)e^{-t}dt<+\infty $ , then $\sum _{n=1}^{+\infty }a_n<+\infty $ . In the following, we will prove $\int _{0}^{+\infty }\tilde c(t)e^{-t}<+\infty $ . By the definition of $\tilde c(t)$ , we have

(33) $$ \begin{align} \int_{0}^{+\infty}\tilde c(t)e^{-t}=\int_{0}^{1}\tilde a(t)dt+\sum_{n=1}^{+\infty}\int_{n}^{n+1}\tilde a(t)dt \leq c(0)e+e\sum_{n=1}^{+\infty}b_n.\end{align} $$

Take $I=\{n_i:n_i$ is the ith positive integer such that $a_{n_i}=b_{n_i}\}\in \mathbb {N}^+$ . Note that if $a_{n+1}\not =b_{n+1}$ , then $b_{n+1}=\frac {b_{n}}{e}$ , thus, we have

(34) $$ \begin{align} \begin{aligned} \sum_{n=1}^{+\infty}b_n=&\sum_{i=1}\sum_{j=0}^{n_{i+1}-n_i-1}b_{n_i+j}\\ =&\sum_{i=1}\sum_{j=0}^{n_{i+1}-n_i-1}b_{n_i}e^{-j}\\ \leq&\sum_{i=1}a_{n_i}\frac{e}{e-1}\\ <&+\infty , \end{aligned} \end{align} $$

where if $n_i$ is the largest integer such that $a_{n_i}=b_{n_i}$ , take $n_{i+1}=+\infty $ . Combining inequality (33) and (34), we obtain $\int _{0}^{+\infty }\tilde c(t)e^{-t}dt<+\infty $ . By replacing $\tilde c(t)$ by $\tilde c(t)+1$ , we have $\tilde {c}\geq c$ , $\tilde {c}$ is increasing on $(1,+\infty )$ , $\tilde c(t)e^{-t}$ is strictly decreasing on $(T,+\infty )$ and $\int _{0}^{+\infty }\tilde {c}(t)e^{-t}dt<+\infty $ .

Moreover, if $\int _{T}^{+\infty }c(t)e^{-t}dt<+\infty $ and $c\in \mathcal {P}_T$ , as $\tilde {c}(t)\geq c(t)$ on $(T,+\infty )$ and $\tilde {c}(t)=ec(t)+1$ on $(T,1)$ , we have $\int _{T}^{+\infty }\tilde {c}(t)e^{-t}dt<+\infty $ and $\tilde c\in \mathcal {P}_T$ . Thus, Lemma 2.10 holds.

Proof. Let w be a coordinate on a neighborhood $V_{z_0}\subset \subset U$ of $z_0$ , such that $w(z_0)=0$ and $G_{\Omega }(z,z_0)=\log |w(z)|+v(w(z))$ , where v is a harmonic function on $V_{z_0}$ and $\sup_{V_{z_0}}|v(w(z))|<+\infty $ . Then there exists $t>0$ such that $\{z\in V_{z_0}:\log |w(z)|+v(w(z))<-t\}\subset \subset V_{z_0}$ .

We claim that $\{z\in \Omega :G_{\Omega }(z,z_0)<-t\}\subset \subset V_{z_0}$ , therefore Lemma 2.12 holds. In fact, set

$$ \begin{align*}\tilde{G}(z) = \left\{ \begin{array}{lcl} G_{\Omega}(z,z_0), & \mbox{if} & z \in V_{z_0}, \\ \max\{G_{\Omega}(z,z_0),-t\}, & \mbox{if} & z\in \Omega\backslash V_{z_0}. \end{array} \right. \end{align*} $$

As $\{z\in V_{z_0}:\log |w(z)|+v(w(z))<-t\}\subset \subset V_{z_0}$ , we know $\tilde G(z)$ is subharmonic on $\Omega $ . Lemma 2.11 tells us $\tilde G(z)\leq G_{\Omega }(z,z_0)$ , therefore $\{z\in \Omega :G_{\Omega }(z,z_0)<-t\}=\{z\in V_{z_1}:G_{\Omega }(z,z_0)<-t\}\subset \subset V_{z_0}$ .

Proof. As $V_1\subset \subset U_1\subset \subset \Omega $ , fixed $z\in \Omega \backslash U_1$ , $G_{\Omega }(z,z_1)$ is harmonic with respect to $z_1$ on a open neighborhood of $\overline {V_1}$ . The Harnack inequality shows that there exists a constant $N>0$ such that

(35) $$ \begin{align} \sup_{z_1\in\overline {V_1}}(-G_{\Omega}(z,z_1))\leq N\inf_{z_1\in\overline {V_1}}(-G_{\Omega}(z,z_1)) \end{align} $$

holds of any $z\in \Omega \backslash U_1$ . As $z_0\in V_1$ , it follows from inequality (35) that

$$ \begin{align*} G_{\Omega}(z,z_1)\ge NG(z,z_0) \end{align*} $$

holds for any $(z,z_1)\in (\Omega \backslash U_1)\times V_1$ .